Theorem tpid2 4735

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
tpid2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
tpid2	⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid2

Step	Hyp	Ref	Expression
1		eqid 2762	. . 3 ⊢ 𝐵 = 𝐵
2	1	3mix2i 1352	. 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
3		tpid2.1	. . 3 ⊢ 𝐵 ∈ V
4	3	eltp 4654	. 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶))
5	2, 4	mpbir 234	1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

This proof depends on syntax axioms:   ∨ w3o 1101   = wceq 1569   ∈ wcel 2142  Vcvv 3454  {ctp 4592

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-sn 4589  df-pr 4591  df-tp 4593

This theorem is used by:  1eltp012  12317  hash3tpb  14539  sgncl  15141  sgnsf  33491  signsw0glem  34949  signsw0g  34952  signswmnd  34953  signswrid  34954  kur14lem7  35712  brtpid2  36222  rabren3dioph  43570  oenord1ex  44070  oenord1  44071  fourierdlem102  46950  fourierdlem114  46962  etransclem48  47024  usgrexmpl1tri  48818  usgrexmpl2nb3  48827  usgrexmpl2nb4  48828  usgrexmpl2nb5  48829